Problem: Simplify the following expression and state the condition under which the simplification is valid. $p = \dfrac{8q^3 + 8q^2 - 576q}{6q^2 - 54q + 48}$
Solution: First factor out the greatest common factors in the numerator and in the denominator. $ p = \dfrac {8q(q^2 + q - 72)} {6(q^2 - 9q + 8)} $ $ p = \dfrac{8q}{6} \cdot \dfrac{q^2 + q - 72}{q^2 - 9q + 8} $ Simplify: $ p = \dfrac{4q}{3} \cdot \dfrac{q^2 + q - 72}{q^2 - 9q + 8}$ Next factor the numerator and denominator. $ p = \dfrac{4q}{3} \cdot \dfrac{(q - 8)(q + 9)}{(q - 8)(q - 1)}$ Assuming $q \neq 8$ , we can cancel the $q - 8$ $ p = \dfrac{4q}{3} \cdot \dfrac{q + 9}{q - 1}$ Therefore: $ p = \dfrac{ 4q(q + 9)}{ 3(q - 1)}$, $q \neq 8$